The estimation of the multivariate normal mean is a fundamental problem, highlighted by the inadmissibility of the MLE for $n\geq 3$ under quadratic loss. While shrinkage and empirical Bayes methods leverage joint structure through geometric reasoning or hierarchical modeling, this paper proposes a class of point estimators derived from the prior-free framework of inferential models. We develop a generalized probability integral transform for independent, non-i.i.d observations, creating a bijective mapping from the sample to an ordered-uniform reference distribution. By combining this bijection with an ordered-uniform predictive random set based on a reweighted Anderson-Darling statistic, we ensure valid and efficient inference that captures the global shape structure revealed by the ordered observations. We further introduce a maximin (bottleneck) criterion for combining multiple plausibility contours. To ensure computability, we develop a sampling-with-replacement surrogate that connects the exact formulation to over-parameterized (g)-modeling. Our approach provides a structural explanation of Stein's paradox, showing that the MLE corresponds to a zero-density point of the joint auxiliary distribution, revealing its implausibility from an auxiliary perspective. Numerical studies show that our estimators are competitive with state-of-the-art auto-modeling methods and outperform classical shrinkage and empirical Bayes methods.

翻译：多元正态均值的估计是一个基本问题，其突出表现在于当维度 $n\geq 3$ 时，在二次损失下最大似然估计（MLE）的不可容许性。尽管收缩方法和经验贝叶斯方法通过几何推理或分层建模利用联合结构，但本文提出了一类源自无先验推断模型框架的点估计。我们针对独立非同分布观测数据发展了一种广义概率积分变换，建立了从样本到有序均匀参考分布的双射映射。通过将该双射与基于重加权安德森-达林统计量的有序均匀预测随机集相结合，我们确保了有效且高效的推断，能够捕捉有序观测所揭示的整体形状结构。我们进一步引入了结合多个似然轮廓的极大极小（瓶颈）准则。为确保可计算性，我们开发了一种带放回抽样的替代方法，将精确公式与过参数化的 (g)-建模相联系。我们的方法为斯坦悖论提供了一种结构性解释，表明最大似然估计对应联合辅助分布的零密度点，从辅助视角揭示其不可信性。数值研究表明，我们的估计器与最先进的自动建模方法具有竞争力，并且优于经典的收缩方法和经验贝叶斯方法。